arXiv subjects
Publications and source records attributed to Mikhail V. Berlinkov.
Let $g \in Σ_n$ be a random mapping, $c>0$, and $H$ be the $c$-crown of $g$ having $r$ roots. Then for any constant $α> 1$, $|H| > αr > 0$ with probability $1 - Θ(n^{-1/2})$. Furthermore, for any fixed constant $j \geq 1$, the height gap between the highest and second-highest $c$-branches is at least $j$ with probability $1 - Θ(n^{-1/2})$.
We study synchronizing partial DFAs, which extend the classical concept of synchronizing complete DFAs and are a special case of synchronizing unambiguous NFAs. A partial DFA is called synchronizing if it has a word (called a \emph{reset word}) whose action brings a non-empty subset of states to a unique state and is undefined for all other states. The class of strongly connected partial DFAs is precisely the class of DFAs recognizing the Kleene star of prefix codes. While in the general case the problem of checking whether a partial DFA is synchronizing is PSPACE-complete, we show that in the strongly connected case, this problem can be efficiently reduced to the same problem for a complete DFA. Using combinatorial, algebraic, and formal languages methods, we develop techniques that relate main synchronization problems for strongly connected partial DFAs to the same problems for complete DFAs. In particular, this includes the Černý and the rank conjectures, the problem of finding a reset word, and upper bounds on the length of the shortest reset words of literal automata of finite prefix codes. We conclude that solving fundamental synchronization problems is equally hard in both models, as an essential improvement of the results for one model implies an improvement for the other.
We prove that a random automaton with $n$ states and any fixed non-singleton alphabet is synchronizing with high probability (modulo an unpublished result about unique highest trees of random graphs). Moreover, we also prove that the convergence rate is exactly $1-Θ(\frac{1}{n})$ as conjectured by [Cameron, 2011] for the most interesting binary alphabet case. Finally, we present a deterministic algorithm which decides whether a given random automaton is synchronizing in linear in $n$ expected time and prove that it is optimal.
Given a subset of states $S$ of a deterministic finite automaton and a word $w$, the preimage is the subset of all states mapped to a state in $S$ by the action of $w$. We study three natural problems concerning words giving certain preimages. The first problem is whether, for a given subset, there exists a word \emph{extending} the subset (giving a larger preimage). The second problem is whether there exists a \emph{totally extending} word (giving the whole set of states as a preimage)---equivalently, whether there exists an \emph{avoiding} word for the complementary subset. The third problem is whether there exists a \emph{resizing} word. We also consider variants where the length of the word is upper bounded, where the size of the given subset is restricted, and where the automaton is strongly connected, synchronizing, or binary. We conclude with a summary of the complexities in all combinations of the cases.
In this paper we address the question of synchronizing random automata in the critical settings of almost-group automata. Group automata are automata where all letters act as permutations on the set of states, and they are not synchronizing (unless they have one state). In almost-group automata, one of the letters acts as a permutation on $n-1$ states, and the others as permutations. We prove that this small change is enough for automata to become synchronizing with high probability. More precisely, we establish that the probability that a strongly connected almost-group automaton is not synchronizing is $\frac{2^{k-1}-1}{n^{2(k-1)}}(1+o(1))$, for a $k$-letter alphabet.
Under the assumption $\mathcal{P} \neq \mathcal{NP}$, we prove that two natural problems from the theory of synchronizing automata cannot be solved in polynomial time. The first problem is to decide whether a given reachable partial automaton is synchronizing. The second one is, given an $n$-state binary complete synchronizing automaton, to compute its reset threshold within performance ratio less than $d \ln{(n)}$ for a specific constant $d>0$.
It is known, that an $ε$-machine is either exactly or asymptotically synchronizing. In the exact case, the observer can infer the current machine state after observing $L$ generated symbols with probability $1-a^L$ where $0 \leq a<1$ is a so-called synchronization rate constant. In the asymptotic case, the probability of the correct prediction the current machine state after observing $L$ generated symbols tends to $1$ exponentially fast as $1-b^L$ for $0<b<1$ and the infimum of such $b$ is a so-called prediction rate constant. Hence the synchronization and prediction rate constants serve as natural measures of synchronization for $ε$-machines. In the present work we show how to approximate these constants in polynomial time in terms of the number of machine states.
We consider the first problem that appears in any application of synchronizing automata, namely, the problem of deciding whether or not a given $n$-state $k$-letter automaton is synchronizing. First we generalize results from \cite{RandSynch},\cite{On2Problems} for the case of strongly connected partial automata. Specifically, for $k>1$ we show that an automaton is synchronizing with probability $1-O(\frac{1}{n^{0.5k}})$ and present an algorithm with linear in $n$ expected time, while the best known algorithm is quadratic on each instance. This results are interesting due to their applications in synchronization of finite state information sources. After that we consider the synchronization of reachable partial automata that has application for splicing systems in computational biology. For this case we prove that the problem of testing a given automaton for synchronization is NP-complete.
In 1964 Černý conjectured that each $n$-state synchronizing automaton posesses a reset word of length at most $(n-1)^2$. From the other side the best known upper bound on the reset length (minimum length of reset words) is cubic in $n$. Thus the main problem here is to prove quadratic (in $n$) upper bounds. Since 1964, this problem has been solved for few special classes of \sa. One of this result is due to Kari \cite{Ka03} for automata with Eulerian digraphs. In this paper we introduce a new approach to prove quadratic upper bounds and explain it in terms of Markov chains and Perron-Frobenius theories. Using this approach we obtain a quadratic upper bound for a generalization of Eulerian automata.